Note on beta elements in homotopy, and an application to the prime three case

Author:
Katsumi Shimomura

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1495-1499

MSC (2010):
Primary 55Q45

Published electronically:
December 8, 2009

MathSciNet review:
2578544

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the sphere spectrum localized at an odd prime . Then we have the first beta element , whose cofiber we denote by . We also consider a generalized Smith-Toda spectrum characterized by . In this note, we show that an element of gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of at the prime three to complete a conjecture of Ravenel's: exists if and only if is not congruent to , or mod .

**1.**Mark Behrens and Satya Pemmaraju,*On the existence of the self map 𝑣⁹₂ on the Smith-Toda complex 𝑉(1) at the prime 3*, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic 𝐾-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 9–49. MR**2066495**, 10.1090/conm/346/06284**2.**Shichirô Oka,*The stable homotopy groups of spheres. II*, Hiroshima Math. J.**2**(1972), 99–161. MR**0322865****3.**Shichirô Oka,*Ring spectra with few cells*, Japan. J. Math. (N.S.)**5**(1979), no. 1, 81–100. MR**614695****4.**Shichirô Oka,*A new family in the stable homotopy groups of spheres*, Hiroshima Math. J.**5**(1975), 87–114. MR**0380791****5.**Shichirô Oka,*A new family in the stable homotopy groups of spheres. II*, Hiroshima Math. J.**6**(1976), no. 2, 331–342. MR**0418096****6.**Shichirô Oka,*Realizing some cyclic 𝐵𝑃_{∗}-modules and applications to stable homotopy of spheres*, Hiroshima Math. J.**7**(1977), no. 2, 427–447. MR**0474290****7.**Douglas C. Ravenel,*Complex cobordism and stable homotopy groups of spheres*, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR**860042****8.**Katsumi Shimomura,*The homotopy groups of the 𝐿₂-localized Toda-Smith spectrum 𝑉(1) at the prime 3*, Trans. Amer. Math. Soc.**349**(1997), no. 5, 1821–1850. MR**1370651**, 10.1090/S0002-9947-97-01710-8**9.**Larry Smith,*On realizing complex bordism modules. IV. Applications to the stable homotopy groups of spheres*, Amer. J. Math.**99**(1977), no. 2, 418–436. MR**0433450****10.**Hirosi Toda,*𝑝-primary components of homotopy groups. IV. Compositions and toric constructions*, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math.**32**(1959), 297–332. MR**0111041****11.**Hirosi Toda,*Algebra of stable homotopy of 𝑍_{𝑝}-spaces and applications*, J. Math. Kyoto Univ.**11**(1971), 197–251. MR**0293631****12.**Hirosi Toda,*On spectra realizing exterior parts of the Steenrod algebra*, Topology**10**(1971), 53–65. MR**0271933**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
55Q45

Retrieve articles in all journals with MSC (2010): 55Q45

Additional Information

**Katsumi Shimomura**

Affiliation:
Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan

Email:
katsumi@kochi-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-09-10190-9

Received by editor(s):
April 19, 2009

Published electronically:
December 8, 2009

Communicated by:
Brooke Shipley

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.